 Cox-de Boor recursion formula

For a nondecreasing spline knot sequence $(t_i)$, initialize the order-one B-splines by $N_{i,1}(x)=1_{[t_i,t_{i+1})}(x)$. For $k\ge 2$, the Cox-de Boor recurrence isA term with zero denominator is assigned zero. For clamped endpoint evaluation, take the appropriate one-sided continuous value rather than applying the half-open order-one convention beyond the final knot.^[ref]

Each order-$k$ B-spline is assembled from two neighboring order-$(k-1)$ B-splines. On their active spans the weights are nonnegative, which proves nonnegativity inductively and keeps the support local. For simple equally spaced spline knots and $k=2$, this produces the triangular hat. Evaluating a whole spline at one point can combine these same local affine steps through De Boor's algorithm.